Comment on “Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond” [J. Chem. Phys. 144, 094109 (2016)]

Abstract

Connections between a “SDE decomposition” to other frameworks constructing landscape in non-equilibrium processes were discussed by Zhou and Li [J. Chem. Phys. 144, 094109 (2016)]. It was speculated that the SDE decomposition would not be generally unique. In this comment, we demonstrate both mathematically and physically that the speculation is incorrect and the uniqueness is guaranteed under appropriate conditions. A few related issues are also clarified, such as the limitation of obtaining potential function from steady state distribution. Current demonstration may lead to a better understanding on the structure and robustness of the decomposition framework.

Response to “Comment on ‘Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond”’ [J. Chem. Phys. 145, 147104 (2016)]

“…4 on uniqueness and their proposed boundary conditions are not valid and proper, we will discuss and clarify some other points raised in the YTA comment.…”

“…4 about the "incorrectness" of ZL paper do not hold. Before they claim the uniqueness result in the YTA comment, more considerations are required and more rigorous and detailed mathematical studies are demanded.…”

“…Recently, Yuan, Tang, and Ao (abbreviated as YTA comment in later text) assert in their comment 4 to ZL paper that ZL's non-uniqueness result for SDE decomposition was "incorrect" because a "counter-example" based on O-U process is available to demonstrate the uniqueness, and the starting point of ZL's paper is not their exact definition of SDE decomposition theory, which already contains the boundary conditions to ensure the uniqueness. In this response, we will illustrate the flaws of such conclusions.…”

“…4 have serious logical mistakes. They speculate that since the existence is proved by ZL, then by "iteratively narrowing down" the boundary conditions, the uniqueness result will be obtained.…”